#8126995

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(3,7)(5,8)(9,10,15,20)(11,19,14,16)(12,13,18,17)(21,24,23,22)(26,31)(27,34)(28,36)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,20,19)(12,21,18,23)(13,22,17,24)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(1,2,4,6)(3,5,7,8)(12,18)(13,17)(21,23)(22,24)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,2)(3,5)(4,6)(7,8)(9,12,15,18)(10,17,20,13)(11,21,14,23)(16,24,19,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,2)(3,5)(4,6)(7,8)(9,13,15,17)(10,18,20,12)(11,22,14,24)(16,23,19,21)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,14,15,11)(10,19,20,16)(12,23,18,21)(13,24,17,22)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034191);